1. A Bayesian Approach to QCD Sum Rules
arXiv: [hep-ph] (to be published in PTP)
ストレンジネス核物理２０１０
KEK
Philipp Gubler (TokyoTech)
Collaborators: Makoto Oka (TokyoTech), Kenji Morita (GSI)

2. Contents QCD Sum Rules and the Maximum Entropy Method
Test of the method in the case of the ρmeson channel
First results of the analysis of charmonium at finite temperature
Conclusions and outlook

3. QCD sum rules
M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147, 385 (1979); B147, 448 (1979).
In this method the properties of the two point correlation function is
fully exploited:
is calculated
“perturbatively”,
using OPE
spectral function
of the operator χ
After the Borel transformation:

4. The basic problem to be solved
given ?
“Kernel”
(but only incomplete and with error)
This is an ill-posed problem.
But, one may have additional information on ρ(ω), which can help to constrain the problem:
- Positivity:
- Asymptotic values:

5. The usual approach s
ρ(s)
The spectral function is usually assumed to be describable by a “pole + continuum” form (ground state + excited states):
This very crude ansatz often works surprisingly well, but…
This ansatz may not always be appropriate.
The dependence of the physical results on sth is often quite large.

6. The Maximum Entropy Method
→ Bayes’ Theorem
likelihood function
prior probability
(Shannon-Jaynes entropy)
Corresponds to ordinary χ2-fitting.
“default model”
M. Jarrel and J.E. Gubernatis, Phys. Rep. 269, 133 (1996).
M.Asakawa, T.Hatsuda and Y.Nakahara, Prog. Part. Nucl. Phys. 46, 459 (2001).

7. A first test case: the ρmeson channel
One of the first and most successful application of QCD sum rules was the analysis of the ρ meson channel.
The “pole + continuum” assumption works well in this case.
Adapted from:
Y. Kwon, M. Procura, and W. Weise, Phys. Rev. C 78, (2008).
e+e- → nπ (n: even)
The experimental knowledge of the spectral function allows us generate realistic mock data.

8. OPE data: We use three parameter sets in our analysis:
(from the Gell-Mann-Oakes-Renner relation)

9. Results: Experiment: mρ= 0.77 GeV Fρ= 0.141 GeV
The position and residue of the ρ-meson are reproduced, but not its width.

10. Application to charmonium (J/ψ) at finite temperature
(in collaboration with M.Oka and K. Morita)
- Prediction of “J/ψ Suppression by Quark-Gluon Plasma Formation”
T. Matsui and H. Satz, Phys. Lett. B 178, 416 (1986). … -
During the last 10 years, a picture has emerged from studies using lattice QCD (and MEM), where J/ψ survives above TC.
(schematic)
M. Asakawa and T. Hatsuda, Phys. Rev. Lett (2004).
S. Datta et al, Phys. Rev. D69, (2004).
T. Umeda et al, Eur. Phys. J. C39S1, 9 (2005). …
taken from
H. Satz, Nucl.Part.Phys. 32, 25 (2006).

11. The charmonium sum rules at T=0
We analyze two different sum rules:
Moment sum rules
Borel sum rules
Developed and analyzed in:
M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147, 385 (1979); B147, 448 (1979).
L.J. Reinders, H.R. Rubinstein and S. Yazaki, Nucl. Phys. B 186, 109 (1981).
R.A. Bertlmann, Nucl. Phys. B 204, 387 (1982).

12. - The J/ψ peak is obtained at the correct value.
Analysis at T=0
Moment sum rules
Borel sum rules
(Moments n=5~14 are used)
mJ/ψ=3.19 GeV
mJ/ψ=3.11 GeV
- The J/ψ peak is obtained at the correct value.
- We also get a (nonsignificant) second peak.
→ Excited states?
→ MEM artifact?

13. The charmonium sum rules at finite T
The application of QCD sum rules has been developed in:
T.Hatsuda, Y.Koike and S.H. Lee, Nucl. Phys. B 394, 221 (1993).
depend on T
A non-scalar twist-2 gluon condensates appears due to the non-existence of Lorentz invariance at finite temperature:
four-velocity of the medium

14. The T-dependence of the condensates
The energy-momentum tensor is considered:
After matching the trace part and the traceless part, one gets:
obtained from lattice QCD
These values are obtained from quenched lattice calculations:
G. Boyd et al, Nucl. Phys. B 469, 419 (1996).
taken from:
O. Kaczmarek et al, Phys. Rev. D 70, (2004).
K. Morita and S.H. Lee, arXiv: [hep-ph].
K. Morita and S.H. Lee, Phys. Rev. Lett. 100, (2008).
K. Morita and S.H. Lee, Phys. Rev. C. 77, (2008).

15. The charmonium spectral function at finite T
Moment sum rules
(Moments n=5~14 are used)
preliminary
melting of J/ψ!

16. The charmonium spectral function at finite T
Borel sum rules
preliminary
melting of J/ψ!

17. Conclusions We have shown that MEM can be applied to QCD sum rules
The “pole + continuum” ansatz is not a necessity
We could observe the melting of J/ψ using finite temperature QCD sum rules and MEM
J/ψ seems to melt between T ~ 1.2 TC and T ~ 1.4 TC,, which is below the values obtained in lattice QCD

18. Outlook (Possible further applications)
Baryonic channels (with K.Ohtani)
Behavior of other hadrons at finite temperature or density
Tetraquarks
Pentaquarks
scattering states ↔ resonances ?